1 A SQUARE AND A CUBE
Queen Ratnamanjuri had a will written that described her fortune of
ratnas (precious stones) and also included a puzzle. Her son Khoisnam
and their 99 relatives were invited to the reading of her will. She wanted
to leave all of her ratnas to her son, but she knew that if she did so, all
their relatives would pester Khoisnam forever. She hoped that she had
taught him everything he needed to know about solving puzzles. She left
the following note in her will—
“I have created a puzzle. If all 100 of you answer it at the same time, you
will share the ratnas equally. However, if you are the first one to solve the
problem, you will get to keep the entire inheritance to yourself. Good luck.”
The minister took Khoisnam and his 99 relatives to a secret room in
the mansion containing 100 lockers.
The minister explained— “Each person is assigned a number from 1 to
100.
• Person 1 opens every locker.
• erson 2 toggles every 2nd locker (i.e., closes it if it is open, opens
P
it if it is closed).
• Person 3 toggles every 3rd locker (3rd, 6th, 9th, … and so on).
• Person 4 toggles every 4th locker (4th, 8th, 12th, … and so on).
This continues until all 100 get their turn.
In the end, only some lockers remain open. The open lockers reveal
the code to the fortune in the safe.”
Before the process begins, Khoisnam realises that he
already knows which lockers will be open at the end.
How did he figure out the answer?
Hint: Find out how many times each locker is toggled.
Reprint 2026-27
, Ganita Prakash | Grade 8
If a locker is toggled an odd number of times, it will be open. Otherwise,
it will be closed. The number of times a locker is toggled is the same as
the number of factors of the locker number. For example, for locker #6,
Person 1 opens it, Person 2 closes it, Person 3 opens it and Person 6 closes
it. The numbers 1, 2, 3, and 6 are factors of 6. If
the number of factors is even, the locker will be 6:
toggled by an even number of people and it will 1×6
eventually be closed. 2×3
Note that each factor of a number has a Factors are
‛partner factor’ so that the product of the pair 1, 2, 3 and 6.
of factors yields the given number. Here, 1 and
6 form a pair of partner factors of 6, and 2 and 3
form another pair.
Does every number have an even number of factors?
4: 9:
1:
1×4 1×9
1×1
2×2 3×3
The only factor
Factors are Factors are
is 1.
1, 2 and 4. 1, 3 and 9.
We see in some cases, like 2 × 2, that the numbers in the pair are the
same.
Can you use this insight to find more numbers with an odd number of
factors?
For instance, 36 has a factor pair 6 × 6 where both numbers are 6.
Does this number have an odd number of factors? If every factor of 36
other than 6 has a different factor as its partner, then we can be sure
that 36 has an odd number of factors. Check if this is true.
Hence all the following numbers have an odd number of factors —
1 × 1, 2 × 2, 3 × 3, 4 × 4, ...
A number that can be expressed as the product of a number with
itself is called a square number, or simply a square. The only numbers
that have an odd number of factors are the squares, because they each
have one factor which, when multiplied by itself, equals the number.
Therefore, every locker whose number is a square will remain open.
2
Reprint 2026-27
, A Square and A Cube
Write the locker numbers that remain open.
Khoisnam immediately collects word clues from these 10 lockers and
reads, “The passcode consists of the first five locker numbers that
were touched exactly twice.”
Which are these five lockers?
The lockers that are toggled twice are the prime numbers, since each
prime number has 1 and the number itself as factors. So, the code is
2-3-5-7-11.
1.1 Square Numbers
Why are the numbers, 1, 4, 9, 16, …, called squares? We know that the
number of unit squares in a square (the area of a square) is the product
of its sides. The table below gives the areas of squares with different
sides.
Sidelength Area
(in units) (in sq units)
1 1 × 1 = 1 sq. unit
2 2 × 2 = 4 sq. units
3 3 × 3 = 9 sq. units
4 4 × 4 = 16 sq. units
5 5 × 5 = 25 sq. units
10 10 × 10 = 100 sq. units
We use the following notation for squares.
1 × 1 = 12 = 1
2 × 2 = 22 = 4
3 × 3 = 32 = 9,
4 × 4 = 42 = 16
5 × 5 =. 52 = 25.
..
In general, for any number n, we write n × n = n2, which is read as ‛n squared’.
Can we have a square of sidelength 35 or 2.5 units?
Yes, there area in square units are ( 35 ) = ( 35 ) × ( 35 ) = (25
9
2
),
2
and (2.5) = (2.5) × (2.5) = 6.25.
3
Reprint 2026-27
Queen Ratnamanjuri had a will written that described her fortune of
ratnas (precious stones) and also included a puzzle. Her son Khoisnam
and their 99 relatives were invited to the reading of her will. She wanted
to leave all of her ratnas to her son, but she knew that if she did so, all
their relatives would pester Khoisnam forever. She hoped that she had
taught him everything he needed to know about solving puzzles. She left
the following note in her will—
“I have created a puzzle. If all 100 of you answer it at the same time, you
will share the ratnas equally. However, if you are the first one to solve the
problem, you will get to keep the entire inheritance to yourself. Good luck.”
The minister took Khoisnam and his 99 relatives to a secret room in
the mansion containing 100 lockers.
The minister explained— “Each person is assigned a number from 1 to
100.
• Person 1 opens every locker.
• erson 2 toggles every 2nd locker (i.e., closes it if it is open, opens
P
it if it is closed).
• Person 3 toggles every 3rd locker (3rd, 6th, 9th, … and so on).
• Person 4 toggles every 4th locker (4th, 8th, 12th, … and so on).
This continues until all 100 get their turn.
In the end, only some lockers remain open. The open lockers reveal
the code to the fortune in the safe.”
Before the process begins, Khoisnam realises that he
already knows which lockers will be open at the end.
How did he figure out the answer?
Hint: Find out how many times each locker is toggled.
Reprint 2026-27
, Ganita Prakash | Grade 8
If a locker is toggled an odd number of times, it will be open. Otherwise,
it will be closed. The number of times a locker is toggled is the same as
the number of factors of the locker number. For example, for locker #6,
Person 1 opens it, Person 2 closes it, Person 3 opens it and Person 6 closes
it. The numbers 1, 2, 3, and 6 are factors of 6. If
the number of factors is even, the locker will be 6:
toggled by an even number of people and it will 1×6
eventually be closed. 2×3
Note that each factor of a number has a Factors are
‛partner factor’ so that the product of the pair 1, 2, 3 and 6.
of factors yields the given number. Here, 1 and
6 form a pair of partner factors of 6, and 2 and 3
form another pair.
Does every number have an even number of factors?
4: 9:
1:
1×4 1×9
1×1
2×2 3×3
The only factor
Factors are Factors are
is 1.
1, 2 and 4. 1, 3 and 9.
We see in some cases, like 2 × 2, that the numbers in the pair are the
same.
Can you use this insight to find more numbers with an odd number of
factors?
For instance, 36 has a factor pair 6 × 6 where both numbers are 6.
Does this number have an odd number of factors? If every factor of 36
other than 6 has a different factor as its partner, then we can be sure
that 36 has an odd number of factors. Check if this is true.
Hence all the following numbers have an odd number of factors —
1 × 1, 2 × 2, 3 × 3, 4 × 4, ...
A number that can be expressed as the product of a number with
itself is called a square number, or simply a square. The only numbers
that have an odd number of factors are the squares, because they each
have one factor which, when multiplied by itself, equals the number.
Therefore, every locker whose number is a square will remain open.
2
Reprint 2026-27
, A Square and A Cube
Write the locker numbers that remain open.
Khoisnam immediately collects word clues from these 10 lockers and
reads, “The passcode consists of the first five locker numbers that
were touched exactly twice.”
Which are these five lockers?
The lockers that are toggled twice are the prime numbers, since each
prime number has 1 and the number itself as factors. So, the code is
2-3-5-7-11.
1.1 Square Numbers
Why are the numbers, 1, 4, 9, 16, …, called squares? We know that the
number of unit squares in a square (the area of a square) is the product
of its sides. The table below gives the areas of squares with different
sides.
Sidelength Area
(in units) (in sq units)
1 1 × 1 = 1 sq. unit
2 2 × 2 = 4 sq. units
3 3 × 3 = 9 sq. units
4 4 × 4 = 16 sq. units
5 5 × 5 = 25 sq. units
10 10 × 10 = 100 sq. units
We use the following notation for squares.
1 × 1 = 12 = 1
2 × 2 = 22 = 4
3 × 3 = 32 = 9,
4 × 4 = 42 = 16
5 × 5 =. 52 = 25.
..
In general, for any number n, we write n × n = n2, which is read as ‛n squared’.
Can we have a square of sidelength 35 or 2.5 units?
Yes, there area in square units are ( 35 ) = ( 35 ) × ( 35 ) = (25
9
2
),
2
and (2.5) = (2.5) × (2.5) = 6.25.
3
Reprint 2026-27
